Hands-on Exercise 8

Calibrating Hedonic Pricing Model for Private Highrise Property with GWR Method

Published

March 6, 2023

Modified

March 6, 2023

1 Setting Up

1.1 Installing and Loading Packages

pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary)
  • olsrr: build OLS and perform diagnostic tests

  • corrplot: multivariate data visualisation and analysis

  • ggpubr:

  • GWmodel: calibrate geographical weighted family of models

  • gtsummary:

1.2 Geospatial Data Wrangling

1.2.1 Importing Data

mpsz = st_read(
  dsn = "data/geospatial",
  layer = "MP14_SUBZONE_WEB_PL"
)
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `D:\bellekwang\IS415\handson_ex\HE8\data\geospatial' using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

1.2.2 CRS Projection

Ensure that the CRS is has the correct ESPG code (3414).

mpsz_svy21 <- st_transform(mpsz, 3414)
st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

1.3 Aspatial Data Wrangling

1.3.1 Importing Data

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")

1.3.2 Conversion to sf Object

condo_resale.sf <- st_as_sf(
  condo_resale,
  coords = c("LONGITUDE", "LATITUDE"),
  crs = 4326
) %>%
  st_transform(crs = 3414)

1.3.3 Final sf Object

head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLI…¹ AREA_…²   AGE PROX_…³ PROX_…⁴ PROX_…⁵ PROX_…⁶ PROX_…⁷ PROX_…⁸
     <dbl>   <dbl>   <dbl> <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>
1   118635 3000000     309    30    7.94   0.166   2.52     6.62   1.77   0.0584
2   288420 3880000     290    32    6.61   0.280   1.93     7.51   0.545  0.616 
3   267833 3325000     248    33    6.90   0.429   0.502    6.46   0.378  0.141 
4   258380 4250000     127     7    4.04   0.395   1.99     4.91   1.68   0.382 
5   467169 1400000     145    28   11.8    0.119   1.12     6.41   0.565  0.461 
6   466472 1320000     139    22   10.3    0.125   0.789    5.09   0.781  0.0994
# … with 12 more variables: PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>, and abbreviated variable names
#   ¹​SELLING_PRICE, ²​AREA_SQM, ³​PROX_CBD, ⁴​PROX_CHILDCARE, ⁵​PROX_ELDERLYCARE,
#   ⁶​PROX_URA_GROWTH_AREA, ⁷​PROX_HAWKER_MARKET, ⁸​PROX_KINDERGARTEN

2 Exploratory Data Analysis

2.1 Statistical Graphics

2.1.1 Selling Price

ggplot(
  data = condo_resale.sf,
  aes(x=`SELLING_PRICE`)
) +
  geom_histogram(
    bins = 20, color = "black", fill = "light blue"
  )

  • There is a right skewed distribution, more condominium units were transacted at relatively lower prices.

To normalise the distribution, use the log transformation and create a new column using the mutate() function.

condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

Plotting the new transformed column:

ggplot(
  data = condo_resale.sf,
  aes(x=`LOG_SELLING_PRICE`)
) +
  geom_histogram(
    bins = 20, color = "black", fill = "light blue"
  )

  • Distribution is relatively less skewed 😄

2.2 Multiple Histogram Plots Distribution of Variables

To draw multiple small histograms, use ggarange() to organise the number of rows and columns.

AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

2.3 Statistical Point Map

tmap_mode("view")
tm_shape(mpsz_svy21)+
  tm_polygons() +
  tmap_options(check.and.fix = TRUE) +
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(10,14))

3 Hedonic Pricing Modelling

3.1 Simple Linear Regression

To build a simple linear regression model, call the lm() function.

  • formula = dependent variable ~ independent variable
condo.slr <- lm(
  formula = SELLING_PRICE ~ AREA_SQM,
  data = condo_resale.sf
)
summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3695815  -391764   -87517   258900 13503875 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515 
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16
  • \(SELLINGPRICE = -258121.1 + 14719x1\)

  • R-sqaured: 0.4518

    • model built is able to explain about 45% of the resale prices
  • p-value << 0.0001

    • reject null hypothesis, mean is a good estimator of selling price

    • slr model is a good estimator

3.1.1 Best Fit Curve

ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm)

  • There are a few statistical outliers with relatively high selling prices

3.2 Multiple Linear Regression

3.2.1 Relationship of Independent Variables

Understanding the multicollinearity of the variables is important before building the MLR model. The independent variables should not be highly correlated to each other.

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")

  • Freehold is highly correlated to LEASE_99YEAR, LEASE_99YEAR will be excluded

3.2.2 Building the MLR Model

condo.mlr <- lm(
  formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
  data=condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3475964  -293923   -23069   241043 12260381 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 ** 
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 ** 
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782    
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .  
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 ** 
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107    
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953    
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 ** 
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 ** 
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474 
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16
  • Looking at the p-values of the independent variables, not all variables are statistically significant.

    • PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_TOP_PRIMARY_SCH, PROX_SUPERMARKET

    • These variables should be removed.

condo.mlr1 <- lm(
  formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
 data=condo_resale.sf)

3.2.3 Preparing Publication Quality Table (olsrr)

Using ols_regress() function, calibrate the revised model.

ols_regress(condo.mlr1)
                             Model Summary                               
------------------------------------------------------------------------
R                       0.807       RMSE                     755957.289 
R-Squared               0.651       Coef. Var                    43.168 
Adj. R-Squared          0.647       MSE                571471422208.591 
Pred R-Squared          0.638       MAE                      414819.628 
------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000 
Residual      8.120609e+14        1421    571471422208.591                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200 
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382 
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740 
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176 
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613 
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510 
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936 
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980 
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677 
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424 
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288 
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082 
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961 
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560 
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821 
-----------------------------------------------------------------------------------------------------------------

3.2.4 Preparing Publication Quality Table (gtsummary)

This method is prettier 🎀

tbl_regression(condo.mlr1, intercept = TRUE)
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1 CI = Confidence Interval

3.2.4.1 Adding Model Statistics

Using add_glance_table() or add_glance_source_note() function to add model statistics into the report table.

tbl_regression(condo.mlr1, 
               intercept = TRUE) %>% 
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, 
                AIC, statistic,
                p.value, sigma))
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957
1 CI = Confidence Interval

3.3 Checking for Multicolinearity

ols_vif_tol(condo.mlr1)
              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759
  • if VIF < 10: no sign of multicollinearity among independent variables

3.3.1 Checking for Non-linearity

MLR model should fufill assumption that linearity and additivity of the relationship between dependent and independent variables.

Plot out the residual vs fitted values to perform linearity assumption test with ols_plot_resid_fit().

ols_plot_resid_fit(condo.mlr1)

  • Most of the data points are scattered around the 0 line, relationship between dependent and independent variables are linear.

3.3.2 Checking for Normality Assumption

3.3.2.1 Via Plotting

ols_plot_resid_hist(condo.mlr1)

  • Residual of MLR model resembles normal distribution.

3.3.2.2 Via p-values

ols_test_normality(condo.mlr1)
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------
  • p-values << 0.05

    • reject the null hypothesis, there is statistical evidence that residual are normally distributed

3.3.3 Checking for Spatial Autocorrelation

First, export the residual values as a data frame.

mlr.output <- as.data.frame(condo.mlr1$residuals)

Combine it with the condo_resale.sf object.

condo_resale.res.sf <- cbind(condo_resale.sf, 
                        condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)

Next, convert condo_resale.sf from a sf data frame into a spatial points data frame.

condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ... 

3.3.3.1 Plotting the Distribution of Residuals

tmap_mode("view")
tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
  • There is a sign of spatial autocorrelation.

3.3.3.2 Moran’s I

To proof that there is sign of spatial autocorrelation, perform the Moran’s I test.

nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
nb_lw <- nb2listw(nb, style = 'W')
lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:  
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance 
    1.438876e-01    -5.487594e-03     3.758259e-05 
  • p-value << 0.05

    • reject null hypothesis that the residuals are randomly distributed
  • Observed Moran I = 0.143 > 0

    • the residual resembles cluster distribution

3.4 GWModel

3.4.1 Fixed Bandwidth GWR Model

3.4.1.1 Computing Fixed Bandwidth

bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.378294e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3408 CV score: 4.721292e+14 
Fixed bandwidth: 971.3403 CV score: 4.721292e+14 
Fixed bandwidth: 971.3406 CV score: 4.721292e+14 
Fixed bandwidth: 971.3404 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
  • The fixed bandwidth should be 971.3405m.

3.4.1.2 Creating the Model

gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                         PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                         PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                         FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale.sp, 
                       bw=bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)
gwr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2023-03-06 09:52:27 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.3405 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3600e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7425e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5000e+06 -1.5970e+05  3.1971e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8073e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112793548
   AREA_SQM                 21575
   AGE                     434201
   PROX_CBD               2704596
   PROX_CHILDCARE         1654087
   PROX_ELDERLYCARE      38867814
   PROX_URA_GROWTH_AREA  78515730
   PROX_MRT               3124316
   PROX_PARK             18122425
   PROX_PRIMARY_SCH       4637503
   PROX_SHOPPING_MALL     1529952
   PROX_BUS_STOP         11342182
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720744
   FREEHOLD               6073636
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3804 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6196 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.53407e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430417 

   ***********************************************************************
   Program stops at: 2023-03-06 09:52:29 
  • AICc of GWR is 42263.1

    • significantly smaller than the global MLR model of 42967.1

3.4.2 Adaptive Bandwidth GWR Model

3.4.2.1 Computing the Adaptive Bandwidth

bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE  + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE    + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL   + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data=condo_resale.sp, 
                      approach="CV", 
                      kernel="gaussian", 
                      adaptive=TRUE, 
                      longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
  • Recommended data points to use: 30

3.4.2.2 Creating the Model

gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale.sp, bw=bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)
gwr.adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2023-03-06 09:52:39 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2023-03-06 09:52:40 
  • AICc of adaptive distance GWR: 41982.22

    • even smaller than AICc of fixed distance GWR of 42263.61

3.4.3 Visualising GWR Output

3.4.3.1 Convert SDF into sf data frame

condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs=3414)
condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
condo_resale.sf.adaptive.svy21  
Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
    Intercept  AREA_SQM        AGE  PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
1   2050011.7  9561.892  -9514.634 -120681.9      319266.92       -393417.79
2   1633128.2 16576.853 -58185.479 -149434.2      441102.18        325188.74
3   3433608.2 13091.861 -26707.386 -259397.8     -120116.82        535855.81
4    234358.9 20730.601 -93308.988 2426853.7      480825.28        314783.72
5   2285804.9  6722.836 -17608.018 -316835.5       90764.78       -137384.61
6  -3568877.4  6039.581 -26535.592  327306.1     -152531.19       -700392.85
7  -2874842.4 16843.575 -59166.727 -983577.2     -177810.50       -122384.02
8   2038086.0  6905.135 -17681.897 -285076.6       70259.40        -96012.78
9   1718478.4  9580.703 -14401.128  105803.4     -657698.02       -123276.00
10  3457054.0 14072.011 -31579.884 -234895.4       79961.45        548581.04
   PROX_URA_GROWTH_AREA    PROX_MRT  PROX_PARK PROX_PRIMARY_SCH
1            -159980.20  -299742.96 -172104.47        242668.03
2            -142290.39 -2510522.23  523379.72       1106830.66
3            -253621.21  -936853.28  209099.85        571462.33
4           -2679297.89 -2039479.50 -759153.26       3127477.21
5             303714.81   -44567.05  -10284.62         30413.56
6             -28051.25   733566.47 1511488.92        320878.23
7            1397676.38 -2745430.34  710114.74       1786570.95
8             269368.71   -14552.99   73533.34         53359.73
9            -361974.72  -476785.32 -132067.59        -40128.92
10           -150024.38 -1503835.53  574155.47        108996.67
   PROX_SHOPPING_MALL PROX_BUS_STOP  NO_Of_UNITS FAMILY_FRIENDLY  FREEHOLD
1          300881.390     1210615.4  104.8290640       -9075.370  303955.6
2          -87693.378     1843587.2 -288.3441183      310074.664  396221.3
3         -126732.712     1411924.9   -9.5532945        5949.746  168821.7
4          -29593.342     7225577.5 -161.3551620     1556178.531 1212515.6
5           -7490.586      677577.0   42.2659674       58986.951  328175.2
6          258583.881     1086012.6 -214.3671271      201992.641  471873.1
7         -384251.210     5094060.5   -0.9212521      359659.512  408871.9
8          -39634.902      735767.1   30.1741069       55602.506  347075.0
9          276718.757     2815772.4  675.1615559      -30453.297  503872.8
10        -454726.822     2123557.0  -21.3044311     -100935.586  213324.6
         y    yhat    residual CV_Score Stud_residual Intercept_SE AREA_SQM_SE
1  3000000 2886532   113468.16        0    0.38207013     516105.5    823.2860
2  3880000 3466801   413198.52        0    1.01433140     488083.5    825.2380
3  3325000 3616527  -291527.20        0   -0.83780678     963711.4    988.2240
4  4250000 5435482 -1185481.63        0   -2.84614670     444185.5    617.4007
5  1400000 1388166    11834.26        0    0.03404453    2119620.6   1376.2778
6  1320000 1516702  -196701.94        0   -0.72065800   28572883.7   2348.0091
7  3410000 3266881   143118.77        0    0.41291992     679546.6    893.5893
8  1420000 1431955   -11955.27        0   -0.03033109    2217773.1   1415.2604
9  2025000 1832799   192200.83        0    0.52018109     814281.8    943.8434
10 2550000 2223364   326635.53        0    1.10559735    2410252.0   1271.4073
      AGE_SE PROX_CBD_SE PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE
1   5889.782    37411.22          319111.1           120633.34
2   6226.916    23615.06          299705.3            84546.69
3   6510.236    56103.77          349128.5           129687.07
4   6010.511   469337.41          304965.2           127150.69
5   8180.361   410644.47          698720.6           327371.55
6  14601.909  5272846.47         1141599.8          1653002.19
7   8970.629   346164.20          530101.1           148598.71
8   8661.309   438035.69          742532.8           399221.05
9  11791.208    89148.35          704630.7           329683.30
10  9941.980   173532.77          500976.2           281876.74
   PROX_URA_GROWTH_AREA_SE PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE
1                 56207.39    185181.3     205499.6            152400.7
2                 76956.50    281133.9     229358.7            165150.7
3                 95774.60    275483.7     314124.3            196662.6
4                470762.12    279877.1     227249.4            240878.9
5                474339.56    363830.0     364580.9            249087.7
6               5496627.21    730453.2    1741712.0            683265.5
7                371692.97    375511.9     297400.9            344602.8
8                517977.91    423155.4     440984.4            261251.2
9                153436.22    285325.4     304998.4            278258.5
10               239182.57    571355.7     599131.8            331284.8
   PROX_SHOPPING_MALL_SE PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE
1               109268.8         600668.6       218.1258           131474.7
2                98906.8         410222.1       208.9410           114989.1
3               119913.3         464156.7       210.9828           146607.2
4               177104.1         562810.8       361.7767           108726.6
5               301032.9         740922.4       299.5034           160663.7
6              2931208.6        1418333.3       602.5571           331727.0
7               249969.5         821236.4       532.1978           129241.2
8               351634.0         775038.4       338.6777           171895.1
9               289872.7         850095.5       439.9037           220223.4
10              265529.7         631399.2       259.0169           189125.5
   FREEHOLD_SE Intercept_TV AREA_SQM_TV     AGE_TV PROX_CBD_TV
1     115954.0    3.9720784   11.614302  -1.615447 -3.22582173
2     130110.0    3.3460017   20.087361  -9.344188 -6.32792021
3     141031.5    3.5629010   13.247868  -4.102368 -4.62353528
4     138239.1    0.5276150   33.577223 -15.524302  5.17080808
5     210641.1    1.0784029    4.884795  -2.152474 -0.77155660
6     374347.3   -0.1249043    2.572214  -1.817269  0.06207388
7     182216.9   -4.2305303   18.849348  -6.595605 -2.84136028
8     216649.4    0.9189786    4.879056  -2.041481 -0.65080678
9     220473.7    2.1104224   10.150733  -1.221345  1.18682383
10    206346.2    1.4343123   11.068059  -3.176418 -1.35360852
   PROX_CHILDCARE_TV PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1         1.00048819          -3.2612693            -2.846248368 -1.61864578
2         1.47178634           3.8462625            -1.848971738 -8.92998600
3        -0.34404755           4.1319138            -2.648105057 -3.40075727
4         1.57665606           2.4756745            -5.691404992 -7.28705261
5         0.12990138          -0.4196596             0.640289855 -0.12249416
6        -0.13361179          -0.4237096            -0.005103357  1.00426206
7        -0.33542751          -0.8235874             3.760298131 -7.31116712
8         0.09462126          -0.2405003             0.520038994 -0.03439159
9        -0.93339393          -0.3739225            -2.359121712 -1.67102293
10        0.15961128           1.9461735            -0.627237944 -2.63204802
   PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV PROX_BUS_STOP_TV
1   -0.83749312           1.5923022            2.75358842        2.0154464
2    2.28192684           6.7019454           -0.88662640        4.4941192
3    0.66565951           2.9058009           -1.05686949        3.0419145
4   -3.34061770          12.9836105           -0.16709578       12.8383775
5   -0.02820944           0.1220998           -0.02488294        0.9145046
6    0.86781794           0.4696245            0.08821750        0.7656963
7    2.38773567           5.1844351           -1.53719231        6.2029165
8    0.16674816           0.2042469           -0.11271635        0.9493299
9   -0.43301073          -0.1442145            0.95462153        3.3123012
10   0.95831249           0.3290120           -1.71252687        3.3632555
   NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV  Local_R2
1     0.480589953        -0.06902748    2.621347 0.8846744
2    -1.380026395         2.69655779    3.045280 0.8899773
3    -0.045279967         0.04058290    1.197050 0.8947007
4    -0.446007570        14.31276425    8.771149 0.9073605
5     0.141120178         0.36714544    1.557983 0.9510057
6    -0.355762335         0.60891234    1.260522 0.9247586
7    -0.001731033         2.78285441    2.243875 0.8310458
8     0.089093858         0.32346758    1.602012 0.9463936
9     1.534793921        -0.13828365    2.285410 0.8380365
10   -0.082251138        -0.53369623    1.033819 0.9080753
                    geometry
1  POINT (22085.12 29951.54)
2   POINT (25656.84 34546.2)
3   POINT (23963.99 32890.8)
4  POINT (27044.28 32319.77)
5  POINT (41042.56 33743.64)
6   POINT (39717.04 32943.1)
7   POINT (28419.1 33513.37)
8  POINT (40763.57 33879.61)
9  POINT (23595.63 28884.78)
10 POINT (24586.56 33194.31)
gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))
glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE       <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1              <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1                   <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1              <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1        <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1      <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1  <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1              <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1             <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1      <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1    <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1         <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1           <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1       <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1              <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1               <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2               <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…

3.4.3.2 Visualising Local R2

tmap_mode("view")
tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

3.4.3.3 Visualising Coefficient Estimates

tmap_mode("view")
AREA_SQM_SE <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)

3.4.3.4 Plotting by URA Planning Region

tm_shape(mpsz_svy21[mpsz_svy21$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)